A "faithful number" is a non-prime number that is one less or one more than some prime number but not both. For example, for numbers up to 20, the numbers 1 8, 10, 14, 16 and 20 are faithful. The number 4 does not qualify because it is equal to "3 + 1" and "5 - 1".
If both 'x' and 'x+2' are faithful but not to the same prime, the pair (x, x+2) is called a faithful pair. So, from 1 to 20 the faithful pairs are (8, 10) and (14, 16). Faithful pairs are scarse and rarer than primes themselves. We can only find 1 faithful pair for numbers 1 to 10, 5 pairs for numbers up to 50 and 8 pairs up to 100.
Let "P" be the set of all faithful pairs from 1 to a given number "n". We define "F" as the set of all p1, p1 < p2 ∀pairs (p1,p2) ∈ P. Write a function "S(n)", that sums all the elements of F.
For 1 to 20, P(20) = [8 10; 14 16], F(20) = [8 14] and S(20) = 22.
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